Prove this identity:

Question

prove the following identity:

2sin^2(x)/sin2(x)+cot(x)=sec(x)csc(x)

 

please note that the cot(x) is not with the sin2(x) as a part of the denominator. The 2sin^2(x)/sin2(x) is its own fraction. And then the cot(x) is added.

Answer 1

2sin²(x)/sin(2x) + cot(x) = 2sin²(x)/(2sin(x)cos(x)) +cos(x)/sin(x)

=sin(x)/cos(x) + cos(x)/sin(x) = sin²(x)/(cos(x)sin(x)) +cos²(x)/(cos(x)sin(x)

= (sin²(x)+cos²(x))/(cos(x)sin(x)) = 1/(cos(x)sin(x)) = sec(x)csc(x)

Answer 2

=2sin^2Ix)/(2 sinx cosx) = sinx/cosx = sinx Sec x = sec x/csc x